Model Independent Greeks

Mathematical Finance Winter School Lunteren

January 2014

Jesper Andreasen Danske Markets, Copenhagen

kwant.daddy@danskebank.com

2

Outline Motivation:  Wots  me  Δelδa?

The short maturity arbitrage condition for implied volatility.

The minimum variance delta.

The ATM MV delta and its relation to the volatility skew.

MV delta in stochastic versus pure local volatility models.

The ATM MV gamma and its relation to the volatility smile.

The ATM MV theta and the term structure of implied volatility.

3

The MV delta when historical and implied parameters are different.

Risk and return in option trading.

Empirical examples.

Conclusion.

4

References Andreasen, J and B Huge (2013):  “Expanding  Forward  Volatility.”  Risk January, 104-107.

Andreasen, J and B Huge (2014):  “Wots  me  Δelδa?”    Unfinished wp, Danske Markets.

Balland, P (2006):  “Forward  Smile.”  Presentation, ICBI Global Derivatives.

Bergomi, L (2004-9):  “Smile  Dynamics  I-IV.” Risk and SSRN.

Dupire, B (2006): “Skew  Modeling.”  Presentation,  ICBI Global Derivatives.

Durrleman,  V  (2004):  ”From  Implied  to  Spot  Volatilities.” PhD dissertation.

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Föllmer,  H  and  D  Sondermann  (1986):  ”Hedging of Non-Redundant Contingent  Claims.”  Contributions to Mathematical Finance, 205-223.

Hagan,  P,  D  Kumar,  A  Lesniewski,  D  Woodward  (2002).  “Managing  Smile  Risk.”  Wilmott Magazine September, 84-108.

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Motivation: Wots me Δelδa? It’s  the most pressing question for any option trader.

The  quant’s  standard  answer is:  “That  depends  on  the  model  – I have ten of them, you pick one...”

The ambiguity comes about because different models specify different dynamics for the implied volatility as the underlying moves.

Let ( , )g g s v be the (normal) option pricing formula and v the corresponding implied volatility. We have

s s v sc g g v

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In a Levy type (jump) model we have ( , ) ( )v s k v k s from which we get

s s v kc g g v

In a local volatility model

1( , ) ( ) ( ) ( )

( )ss k

k

s kv s k O v s k v s ka da

... which leads to the exact opposite of the jump model

,s s v kc g g v k s

Hence, the truly scientific quant would say:

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,s s v kc g g v k s

Hagan et al (2002) argues that the delta can be fine tuned in stochastic local volatility models by changing the correlation versus the local volatility component – when the volatility smile is kept the same.

Dupire (2006), however, counters that and argues that close to ATM, the minimum variance delta is virtually independent of the choice of correlation versus local volatility – when the smile is kept the same.

The minimum variance delta is the position in the underlying stock that (locally) hedges as much variance of the option, i.e. including volatility risk, as possible:

cov[ , ]

var[ ]s vdv dsc c ds

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In this talk we provide a general proof of the Dupire statement in the context of short maturity expansions.

We further show that the MV delta is uniformly higher for low strikes and lower for high strikes in stochastic volatility models than in pure local volatility models.

... and we produce a model free ATM MV gamma, and consider the link between ATM theta and the term structure of implied volatility.

We investigate results empirically and find that there are significant differences  between  “realised”  and  implied  MV  delta.

Ie historical and implied parameters of stochastic volatility models differ.

10

We show how one can create trading strategies that attempt to benefit from this.

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Important Note This is what I knew few months ago.

It turns out, however, that many of our results are well-known to Lorenzo Bergomi, who states the Delta result in his (2004) paper.

Further, our trading strategy ideas are spiritually related to investigations in Bergomi (2009).

It also appears that some of our results can be dug out of Durrleman (2004).

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Short Maturity Expansion Consider the following general class of stochastic volatility models

( , )( , ) ( , )

( , )

ds s z dWdz s z dt s z dZdW dZ s z dt

(1)

Clearly the family of models (1) is rich enough to include several models that fit the same smile.

As an example, one can think of the smile generated by a Heston model but fitted by a pure local volatility model.

Let ( )c t be the time t price of a European option on ( )s T :

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( ) [( ( ) ) ]tc t E s T k (2) Suppose we write the option price as

( ) ( , ( ), ( ))c t g t s t v t (3)

... where ( )g is  Bachelier’s  option  price  formula  and  v is the implied normal volatility. I.e.

( , , ) ( ) ( ) ( ) , ,x x s kg t s v s k v x T tv (4)

We think of v as a stochastic process and we want to identify the conditions v has to satisfy for the option prices to be consistent with absence of arbitrage.

14

Ito expansion of the option price yields

2 21 12 2s v ss sv vvtdc g g ds g dv g ds g ds dv g dv (5)

Using properties of g we obtain

2 21 [ ( ) 2 ] ,2s sss kdc g ds g v dx dt vdv x v (6)

Using that c must be a martingale and therefore [ ] 0tE dc leads to

20 ( ) 2 [ ] ,t

s kdx dt E dv xv v (7) As 0 we get the condition that x needs to be of unit diffusion, i.e.

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2 2 2 2 22 1 , ( ) 0z zs sdx x x x x x s kdt (8)

This is the short maturity arbitrage condition on the implied volatility

ln( / )BS

s ks kv or vx x Equation (8) is the so-called Eikonal equation.

The Eikonal equation is a non-linear first order partial differential equation on the diffusion rather than the linear second order partial differential equations on the drift that we are used to in finance.

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Minimum Variance Delta The MV delta is the position in the underlying stock that minimises the noise of the portfolio of option and stock:

minvar [ ]t dc ds (9)

The idea first appeared in a paper by Föllmer and Sondermann (1986) under the name of locally risk minimizing strategies.

The solution can be found almost directly by rewriting the Brownian motion driver of the volatility process as 2 1/2(1 )dZ dW dB where 0dW dB , i.e.

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2 1/2( )

[ ] (1 ) ( )zs

z zs

dc ds c ds c dz ds O dtc c ds c dB O dt

(10)

This leads to the following break-down of the MV delta:

minvar

[ ]z zs s v sveganaive stickycorrection deltaofdelta strikeforcorr theimplied volatilitydelta

onlydepend onsmile

c c g g v v (11)

So for a given smile, the minimum variance delta of the option price is given from the minimum variance delta of the implied volatility.

...because g and all its derivatives in ( , )s v only depend on the current smile.

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Here, we will identify the ATM MV delta of the implied volatility from the volatility smile.

... and consider what can be said for higher order derivatives.

For later use we define the MV operator

,Df s z f (12)

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Rewriting the PDE Using the relation ( )/x s k v and the MV operator D we can rewrite equation (7) as an equation (directly) in implied volatility

2 2 2 2 2 2 2 2 2 4

3 2 2 2 2 2

0 [ ( ) (1 )( ) ]( ) [2 ( )]( ) [ ]

1 1 12 [ (1 ) ( ( )) ]2 2 2

z

zz zt

Dv v k s v Dv k s v v

v v D v v D v (13)

Equation (13) and differentials of this equation is what we will use for derivation of all results.

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ATM Minimum Variance Delta Differentiating (13) wrt k and evaluating at 0,k s yields

zskv v v Dv (14)

For at-the-money the minimum variance delta of the implied volatility is equal to the slope of the smile.

... for any model without jumps. This is the statement of Bergomi (2004) and Dupire (2006).

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Away from ATM

Using the notation ( ) ( , ; )v k v s z k , we can re-arrange equation (13) as

2 2 2 2 1/24

0

(1 ) ( ) ( )( ) 1( ) { ( ) ( )[1 ] } , 0( ) ( )zv k k sv kDv k v s v kv s s k v k

(15)

As the term inside the square-root is positive for all stochastic volatility models and zero for pure local volatility models we can conclude that...

For a stochastic volatility model relative to a pure local volatility model, the MV Delta is uniformly higher (lower) for k s (k s).

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An example of MV Delta as function of strike in LV and SLV models:

SABR parameters: ( , ) 0.1s z z , 0.5, ( , ) 3s z z . 1s z , 1/12.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0

0.2

0.4

0.6

0.8

1

1.2

0.94 0.96 0.98 1 1.02 1.04 1.06

SABR

LV

v

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ATM Minimum Variance Gamma

Hardcore (Huge) manipulations of (13) lead to

2 2[ ] , 0,kkD v v v k ss z (16)

The ATM MV gamma is determined by the ATM curvature of the smile.

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ATM MV Theta Differentiating (13) with respect to t and T and combining yield

, 0,tt Tv v k s (17)

The ATM theta is determined by the slope of the implied volatility in the maturity dimension.

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Relation to Variance Contracts Using even Huger manipulations of (13) we derive the following relation between the slope of ATM volatility and the variance forward

2

3 2 2 2

var

[ ( ) ]1 1 1 , 0,4 2 2t

T kk k kslopeof contract

E T vv v v v v v v k sT (19)

To apply to VIX, equations need to be converted to BS vols.

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The Difference Between P and Q We often see clear discrepancies between implied and historical parameters.

Before we venture into a discussion of whether this is due to market inefficiency or model misspecification, let us first investigate what it implies for delta hedging.

Suppose the realised dynamics are given by the model as in (1):

( , )( , )

( , )

ds s z dWdz s z dZdW dZ s z dt

(21)

But suppose that options are priced as if they come from the model

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( , )( , )

( , )

ds s z dWdz s z dZdW dZ s z dt

(22)

Consider a portfolio consisting of one option and stocks. The portfolio evolves according to

2 2

2

1 1[ ]2 2[ ] 1

zzss szt

z zs

dc ds c c c c dt

c c dW c dB (23)

Because the pricing model is inconsistent with realised dynamics the portfolio value will not be a martingale, i.e. the hedge strategy will not be mean self-financing.

However, the choice of only affects the risk, not the expected return.

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The delta that minimizes the risk is

[ ]z zs s v sc c g g v v (24)

So the minimum variance delta should be based on the historical (realised) parameter / in combination with derivatives , zsv v computed on implied parameters.

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Estimating MV Delta We have

2

2

( ) 1 ( )

( ) 1

z zs zs

sz

ds v dz O dt v ds v ds dB O dtdv v v

dv v ds O dt ds dBv

(25)

This equation in combination with a model can be used for estimating the historical and the expected noise of the Delta hedge which is proportional to

21

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Realised and Implied MV Delta EUR/USD 1y options over a time series of two years.

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

06-Jun-11 14-Sep-11 23-Dec-11 01-Apr-12 10-Jul-12 18-Oct-12 26-Jan-13 06-May-13 14-Aug-13 22-Nov-13

eta

impl

est

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USD/JPY 1y options over a time series of two years.

Dotted lines are error bars of the statistical error.

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

06-Jun-11 14-Sep-11 23-Dec-11 01-Apr-12 10-Jul-12 18-Oct-12 26-Jan-13 06-May-13 14-Aug-13 22-Nov-13

eta

impl

est

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Significant differences between historical and implied .

…  and  a  lot  of  variation  and co-variation in both quantities.

Which shows

- Implied will not minimize P&L noise.

- Hedge has to be frequently updated.

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Risk and Return The MV delta hedged option position evolves according to (23).

This equation can be re-written using the Eikonal equation (13). We have the ATM limits

2 2 2

1 2 2 2

1{ [ ] 1 ( ) }2

{ [ ] ( ) 1

zssatm delta hedged atm vega riskatm carry

zssk k k kzskew carryatm skew delta hedged

dc ds g dt vv dB O dt

dc ds g v Dv Dv dt v v vv dB ( ) }skew vega risk

O dt

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Which shows that as long as the underlying evolves continuously, the risk of the delta hedged ATM and the skew positions vanish in the short maturity limit.

Using our empirical estimates we can now compute the risk and return in option positions.

So option prices combined with historical estimates actually give both risk and return information.

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Historical Risk and Return Selling ATM 1y USD/JPY volatility (risk and return):

So if your target Sharpe ratio is 1, then sell ATM volatility over first period, then go neutral and buy volatility towards the end of the last  (Abe’nomic)  period.

-200

-150

-100

-50

0

50

100

06-Jun-11 14-Sep-11 23-Dec-11 01-Apr-12 10-Jul-12 18-Oct-12 26-Jan-13 06-May-13 14-Aug-13 22-Nov-13

mu vol

sig vol

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Selling 1y USD/JPY skew (risk and return)

There is no point in either buying or selling skew over the first period, whereas it appears profitable to sell over the last period.

Note that all calculations exclude transaction costs.

-1

0

1

2

3

4

5

6

14-Sep-11 23-Dec-11 01-Apr-12 10-Jul-12 18-Oct-12 26-Jan-13 06-May-13 14-Aug-13 22-Nov-13

mu skew

sig skew

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Model Misspecification?

All results here depend on no jumps.

So the question is whether there are enough jumps to worry.

A simple test for jumps is to consider

2 3 4

varlog

1 1 1ln ( ) ( ) ( )2 3 4contractcontract deltahedge skewcontract kurtosiscontract

s s s ss s s s s

Hence, can we hedge a log-contract with Delta hedge + variance swap?

Or do we need to include skew and kurtosis contracts...?

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This test can be done without any option and interest rate data or assumptions (!) plus we can translate the outcomes in terms of implied volatility, through

21ln ( )/ ( ) ( )2 BSS T S t v T t

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Jumps in FX? Rolling 1y log-contract on USD/JPY data (1971-2013) in implied vol:

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

18.00%

20.00%

0 5 10 15 20 25 30 35 40 45

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Hedging errors:

So the maximum error of hedging log-contract with variance contract over a period of 40y is ~ 0.25% Black volatility!

In other words: the skewness contract never realised more than 0.25% BS vol!

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 5 10 15 20 25 30 35 40 45

(dS/S)^2

(dS/S)^3

(dS/S)^4

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Jumps in Equities? Rolling 1m log-contract on S&P500 (1927-2013) in implied volatility:

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

0 10 20 30 40 50 60 70 80 90 100

d ln S

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Hedging errors in implied volatility terms.

So very little jump risk in selling ATM options. Primary risk is Vega.

-3.0000%

-2.0000%

-1.0000%

0.0000%

1.0000%

2.0000%

3.0000%

4.0000%

5.0000%

6.0000%

0 10 20 30 40 50 60 70 80 90 100

(dS/S)^2

(dS/S)^3

(dS/S)^4

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Conclusion We have produced model free short maturity limits of

- ATM minimum variance delta.

- ATM minimum variance gamma.

We have shown that MV delta is uniformly higher for low strikes and lower for high strikes in stochastic volatility models relative to pure local volatility models.

An ATM theta estimate can be produced from slope in the maturity direction of the implied volatility.

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Results hold for all models without jumps that match the observed smile.

Empirical investigations suggest that historical and implied MV delta can differ significantly.

... and that realised parameters tend to fluctuate more than implied. MV delta should be computed using historical parameter estimates in combination with implied parameters for the option price derivatives.

Differences in implied and historical parameters create non-zero expected returns on option books.

Risk premiums for option positions can be estimated – as opposed to conventional investments.

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Methodologies around the latter need further development.

... for example in the direction of identifying the full efficient frontier of volatility trading.

Qualitatively, our empirical results for FX options are very similar to results obtained for equity options by Bergomi (2004) and Bergomi (2008).